Question

Each side of a box made of metal sheet in cubic shape is 'a' at room temperature 'T', the coefficient of linear expansion of the metal sheet is ' $\alpha^{\prime}$. The metal sheet is heated uniformly, by a small temperature $\Delta T ,$ so that its new temperature is $T +\Delta T$. Calculate the increase in the volume of the metal box.

• A. $3 a ^{3} \alpha \Delta T$
• B. $4 a ^{3} \alpha \Delta T$
• C. $4 \pi a ^{3} \alpha \Delta T$
• D. $\frac{4}{3} \pi a ^{3} \alpha \Delta T$

Answer 1

The Correct Option is A

To determine the increase in the volume of the metal box when it is heated, we begin by understanding the thermal expansion process.

The volume expansion of a solid is determined by its coefficient of linear expansion. For a cube with side length a, the increase in side length when heated by a temperature \Delta T is:

a' = a (1 + \alpha \Delta T)

Here, \alpha is the coefficient of linear expansion. Since the box is a cube, its volume V at room temperature T is:

V = a^3

After heating, the new side length of the cube becomes a', and the new volume V' is:

V' = (a')^3 = (a (1 + \alpha \Delta T))^3

Expanding the above expression using the binomial approximation for small \Delta T, we get:

V' \approx a^3(1 + 3 \alpha \Delta T)

The increase in volume \Delta V is given by:

\Delta V = V' - V = a^3(1 + 3 \alpha \Delta T) - a^3 \Delta V = 3 a^3 \alpha \Delta T

Therefore, the increase in the volume of the metal box when heated by a small temperature \Delta T is:

3 a^3 \alpha \Delta T

This matches the provided correct answer option: 3 a^3 \alpha \Delta T.

The other options can be ruled out because they either involve incorrect coefficients or irrelevant constants, such as \pi, which do not feature in the formula for thermal expansion of a cube.